1 Introduction

2 Methods

Analyses will be conducted in R. All observations from 2010-2013 were removed from analysis due to changes in reporting calculated scores. An additional 33 records with missing values were excluded from overall analysis. Summary statistics for each calculated score will be reviewed, including: number of readmissions, readmission rate, odds of readmission, and log odds. Covariates will be reviewed for sufficient sample size within each group to include in model-fitting. Exploratory data analysis will also include visualizations of empirical log odds (logits) for each calculated score across all potential covariates.
Model fitting will begin with logistic regression considering the association of acuity score with readmission risk. Separate models will be fitted for each acuity score, and expand to consider covariates as appropriate. Additional observations may be dropped or aggregated in consideration of sample size and will be noted in each case. Individual models will be assessed through reviewing deviance residuals and conducting a \(\chi^2\) goodness-of-fit test. Individual predictors will be assessed with confidence intervals. Models will be compared using Akaike’s Information Criteria.

Note: add a MC simulation if there’s enough time!

3 Exploratory Data Analysis

Our primary research interest is whether or not the probability of readmission is associated with calculated score. We calculate the odds \(\frac{p}{1-p}\) empirical log odds \(log(\frac{p}{1-p})\) of readmissions associated with each level of each acuity score and visualize the potential associations, where \(p\) is the probability that a record is of a readmission within 30 days. Summary tables for CCI, LACE, and HOSPITAL can be found in Appendix A: .

A linear association between CCI score and empirical logit is not clear, especially considering score values greater than 8. A linear association between HOSPITAL score and empirical logit is quite pronounced; while the linear association between LACE score and empirical logit is also quite strong, we can observe two potential outliers for LACE score values of 3 and 18.

We also want to consider whether or not probability of readmission changes over time, with respect to acuity score. Various plots examining readmission by score over year can be found in Appendix B; no interaction between score and year is apparent.

We are also interested in the potential effect demographic factors have on the relationship between readmission rate and acuity score. Due to the small sample sizes from considering multiple demographics together, we will disaggregate readmission by each acuity score over one demographic factor (Gender, Race/Ethnicity, Insurance) at a time. Tables and plots for readmission logits by demographic covariates are available in Appendix C. Sample sizes among groups for Race/Ethnicity are too small for this factor to be considered in model fitting; however, previous analysis showed strong evidence that Race/Ethnicity was associated with differences in calculated scores while controlling for other factors. Consequently, this factor should be examined in further study of readmission rates.

4 Model Fitting

Logistic regression is the appropriate model class for our binary response, where we want to model the probability that a record is for a readmission within 30 days. CCI values ranged 0-15; score values of 16, 17 were excluded due to small sample size (n = 1). LACE values ranged 2-19; no values were excluded. HOSPITAL values ranged 0-11; score values of 12 were excluded due to small sample size (n = 2). We start model fitting with a basic model for each calculated score.

4.1 CCI

The base model for readmission against CCI can be written as:

\[ \begin{aligned} log(\frac{p_{CCI_i}}{1-p_{CCI_{i}}}) &= \beta_0 + \beta_1 CCI_i \\ p_{CCI_{i}} &= \frac{e^{\beta_0 + \beta_1CCI_{i}}}{1 + e^{\beta_0 + \beta_1CCI_{i}}} \end{aligned} \] where \(i\ \epsilon\ [0, 14]\)

The estimated odds ratio for CCI is \(e^{\hat{B_1}} = e^{0.003} = 1.003\), 95% CI: (0.98, 1.03); we find no evidence that CCI score is associated with risk for readmission within 30 days.

The estimated model for probability of readmission can be estimated as:

\[p_{CCI_i} = \frac{e^{-1.701 + 0.003*CCI_i}}{1+e^{-1.701 + 0.003*CCI_i}}\] Note: Add manual legend! The line is the fitted model line, and the blue squares are the fitted values. The red circles are the empirical proportions of readmissions for each value of CCI. The black blobs are the raw data for each record. Maybe remove this last piece, since it doesn’t add much to the visual story

Full model diagnostics can be reviewed in Appendix D. No CCI + covariate models need to be fitted since we find no evidence that changes in CCI score are associated with changes in readmission risk.

4.2 LACE

The base model for readmission against LACE can be written as:

\[ \begin{aligned} log(\frac{p_{LACE_i}}{1-p_{LACE_{i}}}) &= \beta_0 + \beta_1 LACE_i \\ p_{LACE_{i}} &= \frac{e^{\beta_0 + \beta_1LACE_{i}}}{1 + e^{\beta_0 + \beta_1LACE_{i}}} \end{aligned} \] where \(i\ \epsilon\ [2, 19]\)

The estimated odds ratio for LACE is \(e^{\hat{B_1}} = e^{0.155} = 1.167\), 95% CI: (1.117, 1.221); these results provide evidence that odds of readmission increases 11.7% - 22.1% for each increased value in LACE score. The confidence intervals given are calculated with an quasibinomial overdispersion parameter \(\phi = 4.26\) to account for extra-binomial variation present in the data. See Appendix E for full model diagnostics.

The estimated model for probability of readmission can be estimated as:

\[p_{LACE_i} = \frac{e^{-3.521 + 0.155*LACE_i}}{1+e^{-3.521 + 0.155*LACE_i}}\]

Note: need to plot base model, fit models with covariates, and perform model selection

4.3 HOSPITAL

The base model for readmission against HOSPITAL can be written as:

\[ \begin{aligned} log(\frac{p_{HOS_i}}{1-p_{HOS_{i}}}) &= \beta_0 + \beta_1 HOS_i \\ p_{HOS_{i}} &= \frac{e^{\beta_0 + \beta_1HOS_{i}}}{1 + e^{\beta_0 + \beta_1HOS_{i}}} \end{aligned} \]
where \(i\ \epsilon\ [0, 11]\)

The estimated odds ratio readmission risk for HOSPITAL is \(e^{\hat{B_1}} = e^{0.251} = 1.285\), 95% CI: (1.253, 1.318); these results provide evidence that odds of readmission increases between 25.3% - 31.8% for each increased value in HOSPITAL score.

The estimated model for probability of readmission can be estimated as:

\[p_{HOSP_i} = \frac{e^{-2.708 + 0.251*HOSP_i}}{1+e^{-2.708 + 0.251*HOSP_i}}\]

Full model diagnostics are available in Appendix F

Note need to plot base model, fit models with covariates, and perform model selection

5 Discussion and Conclusions

Note: will add more based on discussion and additional model fitting

6 Appendices

6.1 Appendix A: Summary Statistics Tables

Table 6.1: Readmission Empirical Statistics by CCI Score
CCI Score Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 41.634 71 10 0.141 0.164 -1.808
1 53.778 243 28 0.115 0.130 -2.038
2 63.191 876 130 0.148 0.174 -1.747
3 71.419 1948 301 0.155 0.183 -1.700
4 82.571 4049 641 0.158 0.188 -1.671
5 81.987 2638 427 0.162 0.193 -1.644
6 83.461 1842 300 0.163 0.195 -1.637
7 84.144 999 141 0.141 0.164 -1.806
8 83.855 517 90 0.174 0.211 -1.557
9 83.144 250 33 0.132 0.152 -1.883
10 82.527 148 21 0.142 0.165 -1.800
11 79.387 75 9 0.120 0.136 -1.992
12 82.533 92 13 0.141 0.165 -1.804
13 84.294 34 11 0.324 0.478 -0.738
14 84.632 19 0 0.000 0.000 -Inf
15 84.857 7 0 0.000 0.000 -Inf
16 79.000 1 0 0.000 0.000 -Inf
17 83.000 1 0 0.000 0.000 -Inf
Table 6.1: Readmission Empirical Statistics by LACE Score
LACE Score Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
2 42.400 5 0 0.000 0.000 -Inf
3 48.200 5 1 0.200 0.250 -1.386
4 57.273 44 1 0.023 0.023 -3.761
5 61.650 100 5 0.050 0.053 -2.944
6 67.569 239 17 0.071 0.077 -2.569
7 69.838 359 44 0.123 0.140 -1.968
8 75.058 652 76 0.117 0.132 -2.025
9 77.822 1115 129 0.116 0.131 -2.034
10 77.225 1462 210 0.144 0.168 -1.785
11 78.390 1683 203 0.121 0.137 -1.987
12 82.185 3043 402 0.132 0.152 -1.882
13 81.588 2997 517 0.173 0.208 -1.568
14 80.683 596 131 0.220 0.282 -1.267
15 79.825 1203 313 0.260 0.352 -1.045
16 80.017 234 80 0.342 0.519 -0.655
17 77.377 53 21 0.396 0.656 -0.421
18 78.273 11 1 0.091 0.100 -2.303
19 81.111 9 4 0.444 0.800 -0.223
Table 6.1: Readmission Empirical Statistics by HOSPITAL Score
HOS Score Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 79.000 86 3 0.035 0.036 -3.320
1 78.553 1198 80 0.067 0.072 -2.637
2 79.491 2045 170 0.083 0.091 -2.401
3 79.933 2578 378 0.147 0.172 -1.761
4 80.051 3934 598 0.152 0.179 -1.719
5 78.902 1585 310 0.196 0.243 -1.414
6 78.322 1532 357 0.233 0.304 -1.191
7 77.354 390 104 0.267 0.364 -1.012
8 75.410 288 92 0.319 0.469 -0.756
9 72.976 124 47 0.379 0.610 -0.494
10 71.750 28 9 0.321 0.474 -0.747
11 71.950 20 7 0.350 0.538 -0.619
12 82.500 2 0 0.000 0.000 -Inf

6.2 Appendix B: Plots of Empirical Logits by Year, Score

6.3 Appendix C: Demographic EDA Tables, Plots

(#tab:appendc_cci)Readmission Empirical Statistics by CCI Score, Gender
CCI Score Gender Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 Female 42.651 43 5 0.116 0.132 -2.028
Male 40.071 28 5 0.179 0.217 -1.526
1 Female 54.348 141 18 0.128 0.146 -1.922
Male 52.990 102 10 0.098 0.109 -2.219
2 Female 63.377 525 68 0.130 0.149 -1.905
Male 62.912 351 62 0.177 0.215 -1.539
3 Female 71.620 1188 165 0.139 0.161 -1.825
Male 71.104 760 136 0.179 0.218 -1.523
4 Female 83.058 2567 352 0.137 0.159 -1.839
Male 81.728 1482 289 0.195 0.242 -1.418
5 Female 82.344 1665 232 0.139 0.162 -1.821
Male 81.376 973 195 0.200 0.251 -1.384
6 Female 83.712 1167 162 0.139 0.161 -1.825
Male 83.028 675 138 0.204 0.257 -1.359
7 Female 84.626 626 78 0.125 0.142 -1.950
Male 83.335 373 63 0.169 0.203 -1.593
8 Female 84.205 317 53 0.167 0.201 -1.606
Male 83.300 200 37 0.185 0.227 -1.483
9 Female 83.339 168 23 0.137 0.159 -1.841
Male 82.744 82 10 0.122 0.139 -1.974
10 Female 83.012 84 6 0.071 0.077 -2.565
Male 81.891 64 15 0.234 0.306 -1.184
11 Female 79.604 48 6 0.125 0.143 -1.946
Male 79.000 27 3 0.111 0.125 -2.079
12 Female 82.621 58 8 0.138 0.160 -1.833
Male 82.382 34 5 0.147 0.172 -1.758
13 Female 85.957 23 8 0.348 0.533 -0.629
Male 80.818 11 3 0.273 0.375 -0.981
14 Female 84.833 12 0 0.000 0.000 -Inf
Male 84.286 7 0 0.000 0.000 -Inf
15 Female 83.333 6 0 0.000 0.000 -Inf
Male 94.000 1 0 0.000 0.000 -Inf
16 Male 79.000 1 0 0.000 0.000 -Inf
17 Female 83.000 1 0 0.000 0.000 -Inf
(#tab:appendc_cci)Readmission Empirical Statistics by CCI Score, Race/Ethnicity
CCI Score Race/Ethnicity Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 African Amer/Black 42.160 25 5 0.200 0.250 -1.386
Asian 37.857 7 1 0.143 0.167 -1.792
Caucasian/White 43.231 26 4 0.154 0.182 -1.705
Hispanic/Latino 41.714 7 0 0.000 0.000 -Inf
Other/Multiracial 36.833 6 0 0.000 0.000 -Inf
1 African Amer/Black 52.292 65 13 0.200 0.250 -1.386
Asian 56.444 9 0 0.000 0.000 -Inf
Caucasian/White 54.760 125 12 0.096 0.106 -2.242
Hispanic/Latino 51.706 17 2 0.118 0.133 -2.015
Other/Multiracial 53.105 19 1 0.053 0.056 -2.890
Unavailable/Unknown 53.500 8 0 0.000 0.000 -Inf
2 African Amer/Black 61.280 164 31 0.189 0.233 -1.456
Asian 63.739 46 7 0.152 0.179 -1.718
Caucasian/White 63.742 565 77 0.136 0.158 -1.847
Hispanic/Latino 62.000 43 6 0.140 0.162 -1.819
Other/Multiracial 63.833 48 7 0.146 0.171 -1.768
Unavailable/Unknown 62.900 10 2 0.200 0.250 -1.386
3 African Amer/Black 69.118 296 47 0.159 0.189 -1.667
Asian 71.280 125 21 0.168 0.202 -1.600
Caucasian/White 72.124 1287 204 0.159 0.188 -1.669
Hispanic/Latino 69.622 82 9 0.110 0.123 -2.093
Other/Multiracial 70.135 111 10 0.090 0.099 -2.313
Unavailable/Unknown 73.149 47 10 0.213 0.270 -1.308
4 African Amer/Black 77.255 404 74 0.183 0.224 -1.495
Asian 79.578 128 14 0.109 0.123 -2.097
Caucasian/White 83.588 3133 497 0.159 0.189 -1.668
Hispanic/Latino 79.292 120 11 0.092 0.101 -2.293
Other/Multiracial 80.909 187 33 0.176 0.214 -1.540
Unavailable/Unknown 83.221 77 12 0.156 0.185 -1.689
5 African Amer/Black 77.605 243 43 0.177 0.215 -1.537
Asian 78.714 91 11 0.121 0.138 -1.984
Caucasian/White 82.993 2034 327 0.161 0.192 -1.653
Hispanic/Latino 79.207 92 12 0.130 0.150 -1.897
Other/Multiracial 79.073 124 19 0.153 0.181 -1.710
Unavailable/Unknown 80.741 54 15 0.278 0.385 -0.956
6 African Amer/Black 80.383 162 24 0.148 0.174 -1.749
Asian 80.385 65 9 0.138 0.161 -1.828
Caucasian/White 84.149 1454 240 0.165 0.198 -1.621
Hispanic/Latino 78.755 53 6 0.113 0.128 -2.058
Other/Multiracial 82.974 76 12 0.158 0.188 -1.674
Unavailable/Unknown 83.031 32 9 0.281 0.391 -0.938
7 African Amer/Black 80.221 95 13 0.137 0.159 -1.842
Asian 80.571 28 3 0.107 0.120 -2.120
Caucasian/White 84.722 809 120 0.148 0.174 -1.748
Hispanic/Latino 80.947 19 0 0.000 0.000 -Inf
Other/Multiracial 87.188 32 2 0.062 0.067 -2.708
Unavailable/Unknown 82.188 16 3 0.188 0.231 -1.466
8 African Amer/Black 79.500 50 8 0.160 0.190 -1.658
Asian 79.417 12 2 0.167 0.200 -1.609
Caucasian/White 84.750 412 72 0.175 0.212 -1.552
Hispanic/Latino 84.385 13 2 0.154 0.182 -1.705
Other/Multiracial 79.962 26 6 0.231 0.300 -1.204
Unavailable/Unknown 83.000 4 0 0.000 0.000 -Inf
9 African Amer/Black 76.522 23 3 0.130 0.150 -1.897
Asian 72.333 6 1 0.167 0.200 -1.609
Caucasian/White 84.710 200 25 0.125 0.143 -1.946
Hispanic/Latino 77.778 9 2 0.222 0.286 -1.253
Other/Multiracial 79.375 8 1 0.125 0.143 -1.946
Unavailable/Unknown 78.750 4 1 0.250 0.333 -1.099
10 African Amer/Black 76.111 9 1 0.111 0.125 -2.079
Asian 78.750 4 0 0.000 0.000 -Inf
Caucasian/White 83.043 115 18 0.157 0.186 -1.684
Hispanic/Latino 81.700 10 0 0.000 0.000 -Inf
Other/Multiracial 85.375 8 2 0.250 0.333 -1.099
Unavailable/Unknown 82.000 2 0 0.000 0.000 -Inf
11 African Amer/Black 79.182 11 2 0.182 0.222 -1.504
Asian 74.000 1 0 0.000 0.000 -Inf
Caucasian/White 80.537 54 4 0.074 0.080 -2.526
Hispanic/Latino 74.000 5 3 0.600 1.500 0.405
Other/Multiracial 68.000 2 0 0.000 0.000 -Inf
Unavailable/Unknown 77.000 2 0 0.000 0.000 -Inf
12 African Amer/Black 81.000 12 0 0.000 0.000 -Inf
Asian 73.000 2 0 0.000 0.000 -Inf
Caucasian/White 83.507 71 10 0.141 0.164 -1.808
Hispanic/Latino 73.333 3 0 0.000 0.000 -Inf
Other/Multiracial 84.000 2 2 1.000 Inf Inf
Unavailable/Unknown 79.000 2 1 0.500 1.000 0.000
13 African Amer/Black 76.000 2 0 0.000 0.000 -Inf
Caucasian/White 84.806 31 10 0.323 0.476 -0.742
Unavailable/Unknown 85.000 1 1 1.000 Inf Inf
14 African Amer/Black 82.500 4 0 0.000 0.000 -Inf
Caucasian/White 85.692 13 0 0.000 0.000 -Inf
Other/Multiracial 82.000 2 0 0.000 0.000 -Inf
15 Asian 66.000 1 0 0.000 0.000 -Inf
Caucasian/White 88.000 6 0 0.000 0.000 -Inf
16 Caucasian/White 79.000 1 0 0.000 0.000 -Inf
17 Caucasian/White 83.000 1 0 0.000 0.000 -Inf
(#tab:appendc_cci)Readmission Empirical Statistics by CCI Score, Insurance
CCI Score Insurance Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 MCARE_MCAID 40.684 19 4 0.211 0.267 -1.322
NO_MCARE_MCAID 41.981 52 6 0.115 0.130 -2.037
1 MCARE_MCAID 54.014 73 9 0.123 0.141 -1.962
NO_MCARE_MCAID 53.676 170 19 0.112 0.126 -2.073
2 MCARE_MCAID 64.725 491 79 0.161 0.192 -1.652
NO_MCARE_MCAID 61.234 385 51 0.132 0.153 -1.879
3 MCARE_MCAID 73.021 1445 235 0.163 0.194 -1.639
NO_MCARE_MCAID 66.815 503 66 0.131 0.151 -1.890
4 MCARE_MCAID 83.836 3455 548 0.159 0.189 -1.669
NO_MCARE_MCAID 75.215 594 93 0.157 0.186 -1.684
5 MCARE_MCAID 82.872 2302 383 0.166 0.200 -1.612
NO_MCARE_MCAID 75.920 336 44 0.131 0.151 -1.893
6 MCARE_MCAID 84.299 1568 255 0.163 0.194 -1.639
NO_MCARE_MCAID 78.668 274 45 0.164 0.197 -1.627
7 MCARE_MCAID 85.226 867 118 0.136 0.158 -1.848
NO_MCARE_MCAID 77.038 132 23 0.174 0.211 -1.556
8 MCARE_MCAID 84.634 462 83 0.180 0.219 -1.519
NO_MCARE_MCAID 77.309 55 7 0.127 0.146 -1.925
9 MCARE_MCAID 84.252 214 28 0.131 0.151 -1.894
NO_MCARE_MCAID 76.556 36 5 0.139 0.161 -1.825
10 MCARE_MCAID 82.919 136 19 0.140 0.162 -1.818
NO_MCARE_MCAID 78.083 12 2 0.167 0.200 -1.609
11 MCARE_MCAID 80.227 66 6 0.091 0.100 -2.303
NO_MCARE_MCAID 73.222 9 3 0.333 0.500 -0.693
12 MCARE_MCAID 82.595 79 12 0.152 0.179 -1.720
NO_MCARE_MCAID 82.154 13 1 0.077 0.083 -2.485
13 MCARE_MCAID 85.452 31 10 0.323 0.476 -0.742
NO_MCARE_MCAID 72.333 3 1 0.333 0.500 -0.693
14 MCARE_MCAID 84.765 17 0 0.000 0.000 -Inf
NO_MCARE_MCAID 83.500 2 0 0.000 0.000 -Inf
15 MCARE_MCAID 88.000 6 0 0.000 0.000 -Inf
NO_MCARE_MCAID 66.000 1 0 0.000 0.000 -Inf
16 MCARE_MCAID 79.000 1 0 0.000 0.000 -Inf
17 MCARE_MCAID 83.000 1 0 0.000 0.000 -Inf

(#tab:appendc_lace)Readmission Empirical Statistics by LACE Score, Gender
LACE Score Gender Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
2 Female 46.750 4 0 0.000 0.000 -Inf
Male 25.000 1 0 0.000 0.000 -Inf
3 Female 48.200 5 1 0.200 0.250 -1.386
4 Female 56.484 31 1 0.032 0.033 -3.401
Male 59.154 13 0 0.000 0.000 -Inf
5 Female 62.148 61 2 0.033 0.034 -3.384
Male 60.872 39 3 0.077 0.083 -2.485
6 Female 67.962 159 8 0.050 0.053 -2.938
Male 66.787 80 9 0.112 0.127 -2.065
7 Female 71.298 242 25 0.103 0.115 -2.161
Male 66.821 117 19 0.162 0.194 -1.641
8 Female 75.782 413 40 0.097 0.107 -2.233
Male 73.808 239 36 0.151 0.177 -1.730
9 Female 78.053 719 69 0.096 0.106 -2.243
Male 77.402 396 60 0.152 0.179 -1.723
10 Female 77.158 931 126 0.135 0.157 -1.855
Male 77.343 531 84 0.158 0.188 -1.672
11 Female 79.351 1106 114 0.103 0.115 -2.164
Male 76.548 577 89 0.154 0.182 -1.702
12 Female 82.776 1970 226 0.115 0.130 -2.043
Male 81.099 1073 176 0.164 0.196 -1.629
13 Female 82.224 1838 292 0.159 0.189 -1.667
Male 80.581 1159 225 0.194 0.241 -1.423
14 Female 81.462 344 69 0.201 0.251 -1.383
Male 79.619 252 62 0.246 0.326 -1.120
15 Female 80.468 656 162 0.247 0.328 -1.115
Male 79.053 547 151 0.276 0.381 -0.964
16 Female 79.297 128 34 0.266 0.362 -1.017
Male 80.887 106 46 0.434 0.767 -0.266
17 Female 77.692 26 12 0.462 0.857 -0.154
Male 77.074 27 9 0.333 0.500 -0.693
18 Female 80.500 2 1 0.500 1.000 0.000
Male 77.778 9 0 0.000 0.000 -Inf
19 Female 78.000 4 2 0.500 1.000 0.000
Male 83.600 5 2 0.400 0.667 -0.405
(#tab:appendc_lace)Readmission Empirical Statistics by LACE Score, Race/Ethnicity
LACE Score Race/Ethnicity Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
2 African Amer/Black 39.000 1 0 0.000 0.000 -Inf
Caucasian/White 56.000 1 0 0.000 0.000 -Inf
Other/Multiracial 39.000 3 0 0.000 0.000 -Inf
3 African Amer/Black 51.000 2 1 0.500 1.000 0.000
Caucasian/White 55.500 2 0 0.000 0.000 -Inf
Other/Multiracial 28.000 1 0 0.000 0.000 -Inf
4 African Amer/Black 50.600 5 1 0.200 0.250 -1.386
Asian 65.500 4 0 0.000 0.000 -Inf
Caucasian/White 57.714 28 0 0.000 0.000 -Inf
Hispanic/Latino 52.000 1 0 0.000 0.000 -Inf
Other/Multiracial 53.000 3 0 0.000 0.000 -Inf
Unavailable/Unknown 59.333 3 0 0.000 0.000 -Inf
5 African Amer/Black 58.769 13 2 0.154 0.182 -1.705
Asian 58.714 7 0 0.000 0.000 -Inf
Caucasian/White 62.864 66 1 0.015 0.015 -4.174
Hispanic/Latino 56.000 6 1 0.167 0.200 -1.609
Other/Multiracial 57.400 5 0 0.000 0.000 -Inf
Unavailable/Unknown 72.667 3 1 0.333 0.500 -0.693
6 African Amer/Black 62.784 37 5 0.135 0.156 -1.856
Asian 68.182 11 0 0.000 0.000 -Inf
Caucasian/White 68.819 166 9 0.054 0.057 -2.859
Hispanic/Latino 66.286 14 1 0.071 0.077 -2.565
Other/Multiracial 64.429 7 1 0.143 0.167 -1.792
Unavailable/Unknown 68.250 4 1 0.250 0.333 -1.099
7 African Amer/Black 65.033 60 8 0.133 0.154 -1.872
Asian 69.611 18 2 0.111 0.125 -2.079
Caucasian/White 70.992 250 33 0.132 0.152 -1.883
Hispanic/Latino 67.083 12 0 0.000 0.000 -Inf
Other/Multiracial 70.750 12 1 0.083 0.091 -2.398
Unavailable/Unknown 73.571 7 0 0.000 0.000 -Inf
8 African Amer/Black 68.493 67 3 0.045 0.047 -3.060
Asian 70.865 37 3 0.081 0.088 -2.428
Caucasian/White 77.197 472 57 0.121 0.137 -1.985
Hispanic/Latino 66.897 29 5 0.172 0.208 -1.569
Other/Multiracial 70.667 33 6 0.182 0.222 -1.504
Unavailable/Unknown 72.714 14 2 0.143 0.167 -1.792
9 African Amer/Black 69.639 108 16 0.148 0.174 -1.749
Asian 76.712 52 5 0.096 0.106 -2.241
Caucasian/White 79.534 839 98 0.117 0.132 -2.023
Hispanic/Latino 68.656 32 2 0.062 0.067 -2.708
Other/Multiracial 74.278 54 6 0.111 0.125 -2.079
Unavailable/Unknown 77.467 30 2 0.067 0.071 -2.639
10 African Amer/Black 69.589 190 28 0.147 0.173 -1.755
Asian 72.017 60 5 0.083 0.091 -2.398
Caucasian/White 79.258 1035 157 0.152 0.179 -1.721
Hispanic/Latino 72.632 68 6 0.088 0.097 -2.335
Other/Multiracial 76.024 85 8 0.094 0.104 -2.264
Unavailable/Unknown 80.292 24 6 0.250 0.333 -1.099
11 African Amer/Black 72.246 187 28 0.150 0.176 -1.737
Asian 73.476 63 11 0.175 0.212 -1.553
Caucasian/White 80.053 1288 148 0.115 0.130 -2.042
Hispanic/Latino 73.186 43 7 0.163 0.194 -1.638
Other/Multiracial 72.827 75 4 0.053 0.056 -2.876
Unavailable/Unknown 76.815 27 5 0.185 0.227 -1.482
12 African Amer/Black 75.956 270 37 0.137 0.159 -1.840
Asian 78.198 106 13 0.123 0.140 -1.968
Caucasian/White 83.314 2379 322 0.135 0.157 -1.854
Hispanic/Latino 78.671 82 5 0.061 0.065 -2.734
Other/Multiracial 80.463 149 13 0.087 0.096 -2.348
Unavailable/Unknown 81.544 57 12 0.211 0.267 -1.322
13 African Amer/Black 76.476 330 56 0.170 0.204 -1.588
Asian 78.133 98 18 0.184 0.225 -1.492
Caucasian/White 82.706 2308 400 0.173 0.210 -1.562
Hispanic/Latino 78.420 100 14 0.140 0.163 -1.815
Other/Multiracial 80.227 110 16 0.145 0.170 -1.771
Unavailable/Unknown 79.882 51 13 0.255 0.342 -1.073
14 African Amer/Black 75.854 82 16 0.195 0.242 -1.417
Asian 73.895 19 4 0.211 0.267 -1.322
Caucasian/White 81.878 417 95 0.228 0.295 -1.221
Hispanic/Latino 78.964 28 2 0.071 0.077 -2.565
Other/Multiracial 80.919 37 10 0.270 0.370 -0.993
Unavailable/Unknown 85.769 13 4 0.308 0.444 -0.811
15 African Amer/Black 76.783 166 44 0.265 0.361 -1.020
Asian 77.465 43 6 0.140 0.162 -1.819
Caucasian/White 80.794 867 226 0.261 0.353 -1.042
Hispanic/Latino 75.455 44 8 0.182 0.222 -1.504
Other/Multiracial 78.683 60 22 0.367 0.579 -0.547
Unavailable/Unknown 81.000 23 7 0.304 0.438 -0.827
16 African Amer/Black 73.467 30 10 0.333 0.500 -0.693
Asian 75.167 6 2 0.333 0.500 -0.693
Caucasian/White 81.606 170 59 0.347 0.532 -0.632
Hispanic/Latino 79.182 11 2 0.182 0.222 -1.504
Other/Multiracial 79.214 14 6 0.429 0.750 -0.288
Unavailable/Unknown 72.000 3 1 0.333 0.500 -0.693
17 African Amer/Black 76.286 14 8 0.571 1.333 0.288
Caucasian/White 79.971 34 11 0.324 0.478 -0.738
Hispanic/Latino 67.500 2 0 0.000 0.000 -Inf
Other/Multiracial 59.667 3 2 0.667 2.000 0.693
18 African Amer/Black 77.000 1 0 0.000 0.000 -Inf
Asian 64.000 1 0 0.000 0.000 -Inf
Caucasian/White 80.000 9 1 0.111 0.125 -2.079
19 African Amer/Black 67.000 2 1 0.500 1.000 0.000
Caucasian/White 84.333 6 3 0.500 1.000 0.000
Hispanic/Latino 90.000 1 0 0.000 0.000 -Inf
(#tab:appendc_lace)Readmission Empirical Statistics by LACE Score, Insurance
LACE Score Insurance Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
2 NO_MCARE_MCAID 42.400 5 0 0.000 0.000 -Inf
3 MCARE_MCAID 49.500 2 1 0.500 1.000 0.000
NO_MCARE_MCAID 47.333 3 0 0.000 0.000 -Inf
4 MCARE_MCAID 63.154 13 0 0.000 0.000 -Inf
NO_MCARE_MCAID 54.806 31 1 0.032 0.033 -3.401
5 MCARE_MCAID 65.851 47 2 0.043 0.044 -3.114
NO_MCARE_MCAID 57.925 53 3 0.057 0.060 -2.813
6 MCARE_MCAID 71.081 160 12 0.075 0.081 -2.512
NO_MCARE_MCAID 60.456 79 5 0.063 0.068 -2.695
7 MCARE_MCAID 73.233 232 32 0.138 0.160 -1.833
NO_MCARE_MCAID 63.638 127 12 0.094 0.104 -2.260
8 MCARE_MCAID 77.881 512 63 0.123 0.140 -1.964
NO_MCARE_MCAID 64.736 140 13 0.093 0.102 -2.279
9 MCARE_MCAID 80.421 878 111 0.126 0.145 -1.933
NO_MCARE_MCAID 68.190 237 18 0.076 0.082 -2.499
10 MCARE_MCAID 79.736 1143 171 0.150 0.176 -1.738
NO_MCARE_MCAID 68.229 319 39 0.122 0.139 -1.971
11 MCARE_MCAID 80.267 1398 172 0.123 0.140 -1.964
NO_MCARE_MCAID 69.182 285 31 0.109 0.122 -2.103
12 MCARE_MCAID 83.816 2585 345 0.133 0.154 -1.871
NO_MCARE_MCAID 72.978 458 57 0.124 0.142 -1.951
13 MCARE_MCAID 82.802 2527 422 0.167 0.200 -1.607
NO_MCARE_MCAID 75.062 470 95 0.202 0.253 -1.373
14 MCARE_MCAID 82.099 517 116 0.224 0.289 -1.240
NO_MCARE_MCAID 71.418 79 15 0.190 0.234 -1.451
15 MCARE_MCAID 81.228 978 257 0.263 0.356 -1.032
NO_MCARE_MCAID 73.724 225 56 0.249 0.331 -1.105
16 MCARE_MCAID 81.346 191 67 0.351 0.540 -0.616
NO_MCARE_MCAID 74.116 43 13 0.302 0.433 -0.836
17 MCARE_MCAID 78.056 36 15 0.417 0.714 -0.336
NO_MCARE_MCAID 75.941 17 6 0.353 0.545 -0.606
18 MCARE_MCAID 82.125 8 1 0.125 0.143 -1.946
NO_MCARE_MCAID 68.000 3 0 0.000 0.000 -Inf
19 MCARE_MCAID 84.667 6 2 0.333 0.500 -0.693
NO_MCARE_MCAID 74.000 3 2 0.667 2.000 0.693

(#tab:appendc_hos)Readmission Empirical Statistics by HOSPITAL Score, Gender
HOSP Score Gender Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 Female 80.000 66 2 0.030 0.031 -3.466
Male 75.700 20 1 0.050 0.053 -2.944
1 Female 78.638 852 47 0.055 0.058 -2.841
Male 78.341 346 33 0.095 0.105 -2.250
2 Female 79.922 1442 103 0.071 0.077 -2.565
Male 78.459 603 67 0.111 0.125 -2.079
3 Female 80.388 1617 207 0.128 0.147 -1.919
Male 79.168 961 171 0.178 0.216 -1.530
4 Female 80.746 2419 334 0.138 0.160 -1.831
Male 78.941 1515 264 0.174 0.211 -1.556
5 Female 79.269 925 165 0.178 0.217 -1.527
Male 78.386 660 145 0.220 0.282 -1.267
6 Female 79.046 849 187 0.220 0.282 -1.264
Male 77.422 683 170 0.249 0.331 -1.104
7 Female 77.465 226 61 0.270 0.370 -0.995
Male 77.201 164 43 0.262 0.355 -1.035
8 Female 74.656 151 47 0.311 0.452 -0.794
Male 76.241 137 45 0.328 0.489 -0.715
9 Female 71.824 68 25 0.368 0.581 -0.542
Male 74.375 56 22 0.393 0.647 -0.435
10 Female 74.385 13 4 0.308 0.444 -0.811
Male 69.467 15 5 0.333 0.500 -0.693
11 Female 69.333 9 2 0.222 0.286 -1.253
Male 74.091 11 5 0.455 0.833 -0.182
12 Female 82.500 2 0 0.000 0.000 -Inf
(#tab:appendc_hos)Readmission Empirical Statistics by HOSPITAL Score, Race/Ethnicity
HOSP Score Race/Ethnicity Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 African Amer/Black 68.857 7 0 0.000 0.000 -Inf
Asian 78.500 8 0 0.000 0.000 -Inf
Caucasian/White 81.127 63 3 0.048 0.050 -2.996
Hispanic/Latino 71.667 3 0 0.000 0.000 -Inf
Other/Multiracial 64.667 3 0 0.000 0.000 -Inf
Unavailable/Unknown 82.000 2 0 0.000 0.000 -Inf
1 African Amer/Black 71.471 102 7 0.069 0.074 -2.608
Asian 76.419 43 4 0.093 0.103 -2.277
Caucasian/White 79.919 939 56 0.060 0.063 -2.758
Hispanic/Latino 72.703 37 2 0.054 0.057 -2.862
Other/Multiracial 74.596 52 7 0.135 0.156 -1.861
Unavailable/Unknown 76.680 25 4 0.160 0.190 -1.658
2 African Amer/Black 72.815 189 13 0.069 0.074 -2.606
Asian 76.269 93 5 0.054 0.057 -2.868
Caucasian/White 80.897 1577 140 0.089 0.097 -2.329
Hispanic/Latino 72.708 65 5 0.077 0.083 -2.485
Other/Multiracial 76.782 87 4 0.046 0.048 -3.033
Unavailable/Unknown 80.118 34 3 0.088 0.097 -2.335
3 African Amer/Black 74.405 259 35 0.135 0.156 -1.856
Asian 76.348 92 18 0.196 0.243 -1.414
Caucasian/White 81.180 1964 291 0.148 0.174 -1.749
Hispanic/Latino 74.867 90 10 0.111 0.125 -2.079
Other/Multiracial 78.620 129 14 0.109 0.122 -2.106
Unavailable/Unknown 78.523 44 10 0.227 0.294 -1.224
4 African Amer/Black 73.988 424 69 0.163 0.194 -1.638
Asian 74.164 152 20 0.132 0.152 -1.887
Caucasian/White 81.576 2969 460 0.155 0.183 -1.696
Hispanic/Latino 75.770 126 10 0.079 0.086 -2.451
Other/Multiracial 77.072 194 27 0.139 0.162 -1.822
Unavailable/Unknown 80.826 69 12 0.174 0.211 -1.558
5 African Amer/Black 73.370 219 40 0.183 0.223 -1.499
Asian 74.840 50 8 0.160 0.190 -1.658
Caucasian/White 80.475 1158 232 0.200 0.251 -1.384
Hispanic/Latino 74.948 58 12 0.207 0.261 -1.344
Other/Multiracial 77.333 72 11 0.153 0.180 -1.713
Unavailable/Unknown 76.571 28 7 0.250 0.333 -1.099
6 African Amer/Black 72.956 225 48 0.213 0.271 -1.305
Asian 74.190 63 11 0.175 0.212 -1.553
Caucasian/White 79.873 1083 259 0.239 0.314 -1.157
Hispanic/Latino 75.037 54 9 0.167 0.200 -1.609
Other/Multiracial 77.603 73 19 0.260 0.352 -1.045
Unavailable/Unknown 78.824 34 11 0.324 0.478 -0.738
7 African Amer/Black 71.172 58 18 0.310 0.450 -0.799
Asian 73.692 13 3 0.231 0.300 -1.204
Caucasian/White 79.707 270 70 0.259 0.350 -1.050
Hispanic/Latino 72.217 23 3 0.130 0.150 -1.897
Other/Multiracial 72.412 17 6 0.353 0.545 -0.606
Unavailable/Unknown 74.333 9 4 0.444 0.800 -0.223
8 African Amer/Black 70.558 43 15 0.349 0.536 -0.624
Asian 74.667 6 0 0.000 0.000 -Inf
Caucasian/White 76.751 205 70 0.341 0.519 -0.657
Hispanic/Latino 74.545 11 1 0.091 0.100 -2.303
Other/Multiracial 68.769 13 3 0.231 0.300 -1.204
Unavailable/Unknown 78.800 10 3 0.300 0.429 -0.847
9 African Amer/Black 66.143 28 16 0.571 1.333 0.288
Asian 68.333 3 0 0.000 0.000 -Inf
Caucasian/White 75.188 80 28 0.350 0.538 -0.619
Hispanic/Latino 88.000 2 0 0.000 0.000 -Inf
Other/Multiracial 69.125 8 3 0.375 0.600 -0.511
Unavailable/Unknown 82.667 3 0 0.000 0.000 -Inf
10 African Amer/Black 76.429 7 1 0.143 0.167 -1.792
Asian 51.000 2 0 0.000 0.000 -Inf
Caucasian/White 71.333 15 6 0.400 0.667 -0.405
Hispanic/Latino 82.000 2 1 0.500 1.000 0.000
Other/Multiracial 61.000 1 1 1.000 Inf Inf
Unavailable/Unknown 77.000 1 0 0.000 0.000 -Inf
11 African Amer/Black 52.750 4 2 0.500 1.000 0.000
Caucasian/White 78.769 13 5 0.385 0.625 -0.470
Hispanic/Latino 64.000 2 0 0.000 0.000 -Inf
Other/Multiracial 76.000 1 0 0.000 0.000 -Inf
12 Caucasian/White 85.000 1 0 0.000 0.000 -Inf
Other/Multiracial 80.000 1 0 0.000 0.000 -Inf
(#tab:appendc_hos)Readmission Empirical Statistics by HOSPITAL Score, Insurance
HOSP Score Insurance Mean Age N Readmit No.  Readmit Prop Readmit Odds Readmit Logit
0 MCARE_MCAID 82.167 72 3 0.042 0.043 -3.135
NO_MCARE_MCAID 62.714 14 0 0.000 0.000 -Inf
1 MCARE_MCAID 81.140 972 71 0.073 0.079 -2.541
NO_MCARE_MCAID 67.425 226 9 0.040 0.041 -3.183
2 MCARE_MCAID 81.553 1705 149 0.087 0.096 -2.346
NO_MCARE_MCAID 69.150 340 21 0.062 0.066 -2.721
3 MCARE_MCAID 81.968 2115 319 0.151 0.178 -1.728
NO_MCARE_MCAID 70.637 463 59 0.127 0.146 -1.924
4 MCARE_MCAID 82.120 3208 498 0.155 0.184 -1.694
NO_MCARE_MCAID 70.905 726 100 0.138 0.160 -1.834
5 MCARE_MCAID 81.006 1267 246 0.194 0.241 -1.423
NO_MCARE_MCAID 70.519 318 64 0.201 0.252 -1.378
6 MCARE_MCAID 80.067 1235 299 0.242 0.319 -1.141
NO_MCARE_MCAID 71.064 297 58 0.195 0.243 -1.416
7 MCARE_MCAID 79.397 307 81 0.264 0.358 -1.026
NO_MCARE_MCAID 69.795 83 23 0.277 0.383 -0.959
8 MCARE_MCAID 77.447 226 77 0.341 0.517 -0.660
NO_MCARE_MCAID 67.984 62 15 0.242 0.319 -1.142
9 MCARE_MCAID 75.382 89 34 0.382 0.618 -0.481
NO_MCARE_MCAID 66.857 35 13 0.371 0.591 -0.526
10 MCARE_MCAID 75.158 19 7 0.368 0.583 -0.539
NO_MCARE_MCAID 64.556 9 2 0.222 0.286 -1.253
11 MCARE_MCAID 74.000 16 5 0.312 0.455 -0.788
NO_MCARE_MCAID 63.750 4 2 0.500 1.000 0.000
12 MCARE_MCAID 82.500 2 0 0.000 0.000 -Inf

6.4 Appendix D: CCI Model Residual Diagnostics

Base CCI model: \(log(\frac{p}{1-p}) = b_0 + b_1*CCI\) :

## 
## Call:
## glm(formula = cbind(n.Readmit, n - n.Readmit) ~ CCI, family = binomial(link = "logit"), 
##     data = readmit_cci_cut)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.5701  -1.2023  -0.5058   0.5187   2.3662  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.70138    0.06099 -27.897   <2e-16 ***
## CCI          0.00285    0.01178   0.242    0.809    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 25.342  on 15  degrees of freedom
## Residual deviance: 25.284  on 14  degrees of freedom
## AIC: 110.25
## 
## Number of Fisher Scoring iterations: 4

The results of the \(\chi^2_{14}\) goodness-of-fit test (p = 0.032) indicate significant evidence of lack of fit. This is likely due to the larger residuals for CCI values 13-14; could refit the model excluding this data which will result in better model fit, but it will not change the practical conclusion reached regarding the association between CCI score and risk of readmission.

6.5 Appendix E: LACE Model Residual Diagnostics

## 
## Call:
## glm(formula = cbind(n.Readmit, n - n.Readmit) ~ LACE, family = binomial(link = "logit"), 
##     data = readmit_lace)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -4.1637  -0.8923   0.7126   1.9487   2.7777  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.52140    0.13552  -25.98   <2e-16 ***
## LACE         0.15467    0.01101   14.04   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 276.245  on 17  degrees of freedom
## Residual deviance:  66.408  on 16  degrees of freedom
## AIC: 158.51
## 
## Number of Fisher Scoring iterations: 4

## 
## Call:
## glm(formula = cbind(n.Readmit, n - n.Readmit) ~ LACE, family = binomial(link = "logit"), 
##     data = readmit_lace_cut)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.1091  -0.6509   0.3533   1.5417   2.4112  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.44691    0.13355  -25.81   <2e-16 ***
## LACE         0.15332    0.01079   14.20   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 258.702  on 16  degrees of freedom
## Residual deviance:  44.266  on 15  degrees of freedom
## AIC: 128.67
## 
## Number of Fisher Scoring iterations: 4

Removing the potential outlier does not improve goodness-of-fit, so rather than remove information we can use a quasibinomial distribution to increase the standard errors of the model estimates and account for the additional variation in the data. Model fitting with covariates may also resolve the issue of extra-binomial variation.

6.6 Appendix F: HOSPITAL Model Residual Diagnostics

## 
## Call:
## glm(formula = cbind(n.Readmit, n - n.Readmit) ~ HOSPITAL, family = binomial(link = "logit"), 
##     data = readmit_hos_cut)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.4851  -1.4034  -0.4645  -0.1126   3.4206  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -2.70804    0.06081  -44.53   <2e-16 ***
## HOSPITAL     0.25061    0.01294   19.37   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 404.380  on 11  degrees of freedom
## Residual deviance:  26.824  on 10  degrees of freedom
## AIC: 101.73
## 
## Number of Fisher Scoring iterations: 4